Chapter 5.5 Limits
In this section, we learn about what it means to take the limit of a function. We will use sequences to make sense of this idea.
Definition and Computation of Limits
Take
View
Goal: Make precise the intuition that
Take
The following statement about
Limit of a function
The real number
Express this symbolically as:
Use this picture to make sense of the idea.
Understand how to take a limit of a function with this example.
Example 1
Take
Take
to be a sequence in in that converges to , Then and , Use the theorems from V.3 to get Since this worked for any sequence in , the conclusion is that .Take
to be a sequence in in that converges to , For , the numerator tends to and the denominator tends to . So the limit laws cannot be used. Rewrite : which is valid for all since Therefore, Since this worked for any sequence in , the conclusion is that
The second limit did not depend on the value of
Verification of the equality
To show that
Namely,
Practice using this definition with this example.
Example 2
Show that the following limits do not exist:
; .
A result from Chapter 5.4 states that a convergent sequence has a unique limit. Therefore, a limit does not exist if there are two different sequences
- Take
. This sequence converges to . Note that implies that Therefore, since , gives us Take . This sequence converges to . Note that implies that Therefore, since , gives us Thus limit does not exist at for
- Notice that if
is a sequence that converges to , then is sequence can diverge to infinity or negative infinity. Take . Notice that is a null sequence and for all . Thus Now take . Notice is a null sequence and for all . Thus limit does not exist at for
It is helpful to establish some initial results on limits of functions.
Limits of Pow Function and Constant Function
For any rational number
, and any at which is defined, .
Understand this theorem with these properties.
Example 3
We have
In order to simplify statements about limits, we introduce the idea of a limit point.
Limit Point
For any subset
If
Understand the idea of a limit point with the figure below.
The limit laws for sequences imply these limit laws for real valued functions on
Limit Laws
For any functions
(limit law for sums)
;(limit law for products)
.(limit law for quotients) If in addition
is not equal to , then .
Use the limit laws to complete the example below.
Example 4
Take
; .
Use the limit laws to get
The limit of the denominator is non-zero. Use the limit laws to get
Use the limit laws to determine the following limits. Pay attention to the difference in the two parts.
Example 5
Determine the following limits and carefully justify your reasoning:
; ;
- Use limit laws to calculate the limit:
- The quotient limit law cannot be used because the limit of the denominator is zero. However, the limit of the numerator is also zero. Rewrite the expression to expose the common zero. Notice that for
Because the definition of the limit restricts to sequences for which , rewrite the expression to get
We have already shown that if
Since these limits are valid for any null sequence with nonzero terms, we have the following.
Special Limits of Sine and Cosine
All other limits involving trigonometric functions can be determined from these limits.
In particular, if
and
Since
Sine, Cosine, and Tangent Limits
For any real number
, so long as is not an odd multiple of
Use the above theorem to compute the following limits.
Example 6
Determine the following limits and carefully justify your reasoning:
;
A result from Chapter 5.4 states that for any null sequence
The quotient limit law cannot be used in both parts because the limit of the denominator is zero. However, the limit of the numerator is also zero. Rewrite the expression to expose the common zero.
Rewrite the expression in order to use the limit laws:
Notice that
Use this and the limit laws to calculate the limit:
The squeeze theorem for sequences implies an analogous theorem for functions that has the same name.
Squeeze Theorem
Squeeze Theorem: For any functions
Practice with the next example.
Example 7
Use the squeeze theorem to determine the following limit:
The limit laws for products does not apply because
Since
for any real number , for any real number . Since is always positive if , what follows is true : Take and . The above calculations demonstrate that for all .Use the limit laws to get
and .
By squeeze theorem,
In the next example, we look at a limit of a composite function.
Example 8
Take
Note that for all
Therefore,
Therefore,
In a future section, we will learn about a limit law for composite functions.
One Sided Limits
The function
Although
This example motivates a refinement of the definition of a limit.
Limit from the Left and Limit from the Right
To say that the real number
- For any sequence
in that converges to , converges to . Express this symbolically as: .
To say that the real number
- For any sequence
in that converges to , converges to . Express this symbolically as: .
Understand the difference with the figure below.
The limits defined above are one sided limits.
All theorems about one sided limits (for example: limit laws, squeeze theorem) are also valid for one sided limits because one sided limits are limits for the function with domain restricted to the left or right of the limit point
Limit Exist if and only if One Sided Limit Exist
Note:
Understand the theorem by using it in the next example.
Example 9
Calculate the following limits and carefully justify your reasoning:
The limit laws do not apply because the limit of the denominator is zero. However, because the limit of the numerator is also zero, rewrite the expression so that the limit laws can apply.
- Take the sequence
in that converges to . Rewrite : which holds for all since . Thus Since this worked for any sequence in , the conclusion is
- Take the sequence
in that converges to . Rewrite . Notice that or Since in , this means for all . Thus which holds for all since . Thus Since this holds for all sequences that converge to , the conclusion is
- Take the sequence
in that converges to . Rewrite . Notice that or Since in , this means for all . Thus which holds for all since . Thus Since this holds for all sequences that converge to , the conclusion is
Use graphical information to compute the following limits.
Example 10
Use the sketch of the function
undefined | ||||
does not exist | does not exist | |||
does not exist | does not exist | |||
does not exist |
Infinite Limits
Take
The following statement about
Limit at Infinity
Take
For any sequence
The real number
is the limit of as tends to .Express this symbolically as:
.
This limit is referred to as a limit at infinity.
Understand the definition with the figure below.
Determine the limit at infinity for the following function.
Example 11
Calculate
Take
Therefore,
Since this holds for any sequence
Therefore,
Since this holds for any sequence
Take
The following statement about
Limit at Negative Infinity
Take
The real number
is the limit of as tends to .Express this symbolically as:
This limit is referred to as a limit at negative infinity.
Use the figure below to understand the meaning of limit at negative infinity.
Calculate the limit at negative infinity of the following example.
Example 12
Calculate
Take
The limit of a function
The following statement about
Diverges to Infinity or Negative Infinity at a Real Number
For any sequence
- Express this symbolically as:
.
Note: The
Understand divergence to infinity or negative infinity with the figures below.
Determine how the limits diverge with the next example.
Example 13
Determine the way in which the following limits diverge:
.
- Take
to be a null sequence for which . Then is positive null sequence, so . Since this is true for any null sequence for which ,
- Take
to be a sequence for which and Notice that so If , then If , then when . Therefore, as tends to , . Since this is true for any sequence for which and , the conclusion is
The divergence to infinity or negative infinity can also happen with one sided limits.
Diverges to Infinity or Negative Infinity to the Left and to the Right
Write
- For any sequence
in that converges to , diverges to .
Write
- For any sequence
in that converges to , diverges to .
Understand the meaning of this with definition visually with this figure.
Practice with this example.
Example 14
Determine the way in which the following limits diverge:
Take
to be a sequence in for which and Since , this means . And since implies that is a null sequence, we have that . ThusTake
to be a sequence in for which and Since , this means . And since implies that is a null sequence, we have that . ThusSince
and this means the limit at does not exist.Follow a similar argument in a,b, and c, to conclude
and , which means that the limit does not exist.
Limits of functions at
Diverges to Infinity or Negative Infinity at Infinity
For any function
- For any sequence
in that diverges to , diverges to .
For any function
- For any sequence
in that diverges to , diverges to .
Understand this type of divergence with the figures below.
Here are two simple examples.
Example 15
We have the following:
The limit laws and squeeze theorem are also valid for convergent limits at infinity, and the proofs are essentially unchanged. These computational tools are very useful for determining limits at infinity and negative infinity. To determine that a limit is an infinite limit typically requires making an estimate.
Example 16
Compute the following limits and carefully justify your reasoning:
Notice that
Since diverges to , the denominator approaches 1 while the numerator diverges to ThusNotice that
Since diverges to , denominator approaches 5 while the numerator diverges to ThusNotice that
Since tends to , the denominator approaches 1 while the numerator diverges to ThusNotice that
Since tends to , the numerator tends to while the denominator tends to . Thus,Use the squeeze theorem. Since the cosine function is bounded between
and , we have that As long as , is defined and positive, so implies that for . Since and , use the squeeze theorem to conclude
In the next example pay attention to how we rewrite the term to obtain the limit.
Example 17
Compute
Rewrite the expression:
The ideas from Chapter 3 and Chapter 4, in regards to horizontal and vertical asymptotes, can be made precise using the limits.
Here is the definition of a horizontal asymptote.
Horizontal Asymptote
The horizontal line that passes through
Understand a horizontal asymptote with the following figure.
This is the definition of a vertical asymptote.
Vertical Asymptote
The vertical line that passes through
Understand this definition with the figure below
Find vertical and horizontal asymptotes with the following example.
Example 18
Determine all horizontal and vertical asymptotes of the function
Horizontal asymptotes can be found by calculating the limit as
The vertical asymptotes of the function
When
To summarize,
and are horizontal asymptotes and are vertical asymptotes.
Asymptotic behavior from Chapter 3 and 4 can also be restated using limits as well.
Asymptotic Behavior
Functions
When will a rational function have the same asymptotic behavior as
Example 19
Take
The leading term of the numerator is
Notice that [===,] Multiply numerator and denominator by
Knowing the asymptotic properties of an invertible function will determine the asymptotic properties of the function’s inverse.
It is helpful to first recognize the following facts:
Special Exponential and Tangent limits
Use these facts to understand how to compute the following limits.
Example 20
Determine the following limits:
; ; ; .
Recall that for an invertible function
; ;
Limits and Paths
Everything we learned about taking limits of functions extends to the idea of taking limits of paths. Here is the definition.
Limit of a Path
For any
- (Condition 1)
- (Condition 2)
and
Practice calculating limits of paths with the following example. When appropriate remember our limit laws and techniques for computing limits.
Example 21
Take
Determine the following limits:
In this problem, the component functions are
Note that
and Therefore,Use a similar argument from above to obtain that
Use the squeeze theorem to obtain that
Also notice that Therefore,Use the squeeze theorem to obtain that
Also notice that Therefore,
Using what we know about limits can help us construct paths with special properties.
Example 22
Identify a path that describes the position in time of a particle that moves along the line segment
There are many different answers. Here is one example.
First the vector that moves points along this line:
The midpoint of the path is
The path of the particle
where
We know that
The path is thus